Exploring Data

The histogram below shows the distribution of the per cent of women aged 15 and over who have never married in each of the 50 states and the District of Columbia.

The center of this distribution is in the interval 

(a) $22\%$ to $24\%$ 

(b) $24\%$ to $26\%$

(c)  $26\% to 28\%$

(d) $28\%$ to $30\%$

(e)$36\%$ to $38\%$

The histogram below shows the distribution of the percent of women aged $15$ and over who have never married in each of the $50$ states and the District of Columbia. 

In about what percent of states have at least $30\%$ of women aged $15$ and over never married?

(a) $4\%$ (b) $7\%$ (c)$10\%$(d) $14\%$ (e) $32\%$

Baseball players (Introduction) Here is a small part of a data set that describes Major League Baseball players as of opening day of the 2009 season: 

(a) What individuals does this data set describe?

(b) In addition to the player’s name, how many variables does the data set contain? Which of these variables are categorical and which are quantitative?

(c) What do you think are the units of measurement for each of the quantitative variables?

The rating service Arbitron asked adults who used several high-tech devices and services whether they “loved” using them. Below is a graph of the percent who said they did.

(a) Summarize what this graph tells you in a sentence or two.

(b) Would it be appropriate to make a pie chart of these data? Why or why not?

A study in Sweden looked at former elite soccer players, people who had played soccer but not at the elite level, and people of the same age who did not play soccer. Here is a two-way table that classifies these individuals by whether or not they had arthritis of the hip or knee by their mid-fifties: 

(a) What percent of the people in this study were elite soccer players? What percent had arthritis? 

(b) What percent of the elite soccer players had arthritis? What percent of those who had arthritis were elite soccer players? 

Refer to Exercise $77$. We suspect that the more serious soccer players have more arthritis later in life. Do the data confirm this suspicion? Give graphical and numerical evidence to support your answer. 

Here, once again, is the stem plot of travel times to work for $20$ randomly selected New Yorkers. Earlier, we found that the median was $22.5$ minutes.

Based only on the stem plot, would you expect the mean travel time to be less than, about the same as, or larger than the median? Why?

Here, once again, is the stem plot of travel times to work for $20$ randomly selected New Yorkers. Earlier, we found that the median was $22.5$

Use your calculator to find the mean travel time. Was your answer to Question $1$ correct? 

Here, once again, is the stem plot of travel times to work for $20$ randomly selected New Yorkers. Earlier, we found that the median was $22.5$ minutes.

Interpret your result from Question 2 in context without using the words “mean” or “average".  

Here, once again, is the stem plot of travel times to work for $20$ randomly selected New Yorkers. Earlier, we found that the median was $22.5$ minutes.

Would the mean or the median be a more appropriate summary of the center of this distribution of drive times? Justify your answer. 

Based only on the stemplot, would you expect the mean travel time to be less than, about the same as, or larger than the median? Why? 

The $2009$ roster of the Dallas Cowboys professional football team included $10$ offensive linemen. Their weights (in pounds) were 

Find the five-number summary for these data by hand. Show your work. 

The $2009$ roster of the Dallas Cowboys professional football team included $10$ offensive linemen. Their weights (in pounds) were 

Calculate the IQR. Interpret this value in the context

The $2009$ roster of the Dallas Cowboys professional football team included 10 offensive linemen. Their weights (in pounds) were 

$338, 318, 353, 313, 318, 326, 307, 317, 311, 311$

Determine whether there are any outliers using the$1.5 \times \mathrm{IQR}$ rule.

The $2009$ roster of the Dallas Cowboys professional football team included $10$ offensive linemen. Their weights (in pounds) were $338 318 353 313 318 326 307 317 311 311$

Draw a boxplot of the data. 

The heights (in inches) of the five starters on a basketball team are $67, 72, 76, 76, \mathrm{and} 84$. Find the interpretation of mean. 

The heights (in inches) of the five starters on a basketball team are $67, 72, 76, 76, \mathrm{and} 84.$

Make a table that shows, for each value, its deviation from the mean and its squared deviation from the mean. 

The heights (in inches) of the five starters on a basketball team are $67, 72, 76, 76, \mathrm{and} 84.$

Show how to calculate the variance and standard deviation from the values in your table. 

The heights (in inches) of the five starters on a basketball team are $67, 72, 76, 76, \mathrm{and} 84.$

Interpret the meaning of the standard deviation in the given setting.

Quiz grades Joey’s first $14$ quiz grades in a marking period were 

Calculate the mean. Show your work. Interpret your result in context. 

Cowboys the $2009$ roster of the Dallas Cowboys football team included $7$ defensive linemen. Their weights (in pounds) were $306, 305, 315, 303, 318, 309, and 285$

Calculate the mean. Show your work. Interpret your result in context. 

Refer to Exercise $79$. 

(a) Find the median by hand. Show your work. Interpret your result in context. 

(b) Suppose Joey has an unexcused absence for the ${15}^{th}$ quiz, and he receives a score of zero. Recalculate the mean and the median. What property of measures of center does this illustrate? 

Cowboys Refer to Exercise $80$

(a) Find the median by hand. Show your work. Interpret your result in context. 

(b) Suppose the lightest lineman had weighed $265$ pounds instead of $285$ pounds. How would this change affect the mean and the median? What property of measures of center does this illustrate? 

Incomes of college grads According to the Census Bureau, the mean and median $2008$ income of people at least $25$ years old who had a bachelor’s degree but no higher degree were $\backslash (48,097$ and $\backslash )60,954$ Which of these numbers is the mean and which is the median? Explain your reasoning. 

House prices: The mean and median selling prices of existing single-family homes sold in November $2009$ were $\backslash (216,400$ and $\backslash )172,600$. Which of these numbers is the mean and which is the median? Explain how you know.

Suppose that a Major League Baseball team’s mean yearly salary for its players is $1.2 million and that the team has 25 players on its active roster. What is the team’s total annual payroll?

If you knew only the median salary, would you be able to answer this question? Why or why not?

Mean salary? Last year a small accounting firm paid each of its five clerks $\backslash (22,000$, two junior accountants $\backslash )50,000$ each, and the firm’s owner $\$270,000$. What is the mean salary paid at this firm? How many of the employees earn less than the mean? What is the median salary? Write a sentence to describe how an unethical recruiter could use statistics to mislead

prospective employees.

Domain names When it comes to Internet domain names, is shorter better? According to one ranking of Web sites in $2008$, the top $8$ sites (by number of

“hits”) were yahoo.com, google.com, youtube.com, live.com, msn.com, myspace.com, wikipedia.org, and facebook.com. These familiar sites certainly have short domain names. The histogram below shows the domain name lengths (in a number of letters in the name, not including the extensions .com and .org) for the $500$ most popular Web sites. 

(a) Estimate the mean and median of the distribution. Explain your method clearly.

(b) If you wanted to argue that shorter domain names were more popular, which measure of the center would you choose—the mean or the median? Justify your answer.

We all know that fruit is good for us. Below is a histogram of the number of servings of fruit per day claimed by $74$ seventeen-year-old girls in a study in Pennsylvania?

(a) With a little care, you can find the median and the quartiles from the histogram. What are these numbers? How did you find them?

(b) Estimate the mean of the distribution. Explain your method clearly.

Refer to Exercise 79 

(a) Find and interpret the interquartile range (IQR)

(b) Determine whether there are any outliers. Show your work

Cowboys Refer to Exercise $80$

(a) Find and interpret the interquartile range (IQR). 

(b) Determine whether there are any outliers. Show your work.

Don’t call me In a September $28, 2008$, article titled “Letting Our Fingers Do the Talking,” the New York Times reported that Americans now send more text messages than they make phone calls. According to a study by Nielsen Mobile, “Teenagers ages $13$ to $17$ are by far the most prolific texters, sending or receiving $1,742$ messages a month.” Mr. Williams, a high school statistics teacher, was skeptical about the claims in the article. So he collected data from his first-period statistics class on the number of text messages and calls they had sent or received in the past $24$ hours. Here are the texting data:

(a) Make a boxplot of these data by hand. Be sure to check for outliers.

(b) Do these data support the claim in the article about the number of texts sent by teens? Justify your answer with appropriate evidence.

Acing the first test Here are the scores of Mrs. Liao’s students on their first statistics test:

(a) Make a boxplot of the test score data by hand. Be sure to check for outliers.

(b) How did the students do on Mrs. Liao’s first test? Justify your answer.

Texts or calls? Refer to Exercise 91. A boxplot of the difference (texts – calls) in the number of texts and calls for each student is shown below. 

(a) Do these data support the claim in the article about texting versus calling? Justify your answer with appropriate evidence.

(b) Can we draw any conclusion about the preferences of all students in the school based on the data from Mr. Williams’s statistics class? Why or

why not?

Electoral votes To become president of the United States, a candidate does not have to receive a majority of the popular vote. The candidate does have to win a majority of the $538$ electoral votes that are cast in the Electoral College. Here is a stemplot of the number of electoral votes for each of the $50$ states and the District of Columbia. 

(a) Make a boxplot of these data by hand. Be sure to check for outliers.

(b) Which measure of center and spread would you use to summarize the distribution—the mean and standard deviation or the median and IQR? Justify your answer.

Comparing investments Should you put your money into a fund that buys stocks or a fund that invests in real estate? The boxplots compare the daily returns (in percent) on a “total stock market” fund and a real estate fund over a year ending in November$2007$

(a) Read the graph: about what were the highest and lowest daily returns on the stock fund? 

(b) Read the graph: the median return was about the same on both investments. About what was the median return? 

(c) What is the most important difference between the two distributions? 

Income and education level Each March, the Bureau of Labor Statistics compiles an Annual Demographic Supplement to its monthly Current Population Survey.44 Data on about $71,067$ individuals between the ages of data-custom-editor="chemistry" $25 \mathrm{and} 64$ who were employed full-time were collected in one of these surveys. The boxplots below compare the distributions of income for people with five levels of education. This figure is a variation of the boxplot idea: because large data sets often contain very extreme observations, we omitted the individuals in each category with the top $5\%$and bottom $5\%$ of incomes. Write a brief description of how the distribution of income changes with the highest level of education reached. Give specifics from the graphs to support your statements.

The level of various substances in the blood influences our health. Here are measurements of the level of phosphate in the blood of a patient, in milligrams of phosphate per deciliter of blood, made on $6$consecutive visits to a clinic: $5.6, 5.2, 4.6, 4.9, 5.7 \mathrm{and} 6.4$. A graph of only $6$ observations gives little information, so we proceed to compute the mean and standard deviation.

(a) Find the standard deviation from its definition. That is, find the deviations of each observation from the mean, square the deviations, and then obtain the variance and the standard deviation.

(b) Interpret the value of six you obtained in (a)

Phosphate levels The level of various substances in the blood influences our health. Here are measurements of the level of phosphate in the blood of a patient, in milligrams of phosphate per deciliter of blood, made on 6 consecutive visits to a clinic: 5.6, 5.2, 4.6, 4.9, 5.7, 6.4. A graph of only 6 observations gives little information, so we proceed to compute the mean and standard deviation. (a) Find the standard deviation from its definition. That is, find the deviations of each observation from the mean, square the deviations, then obtain the variance and the standard deviation. (b) Interpret the value of sx you obtained in (a) 

Feeling sleepy? The first four students to arrive for a first-period statistics class were asked how much sleep (to the nearest hour) they got last night. Their responses were $7, 7, 9,$and $9$

 (a) Find the standard deviation from its definition. That is, find the deviations of each observation from the mean, square the deviations, then obtain the variance and the standard deviation. 

(b) Interpret the value of $s_x$ you obtained in (a). 

(c) Do you think it’s safe to conclude that the mean amount of sleep for all $30$ students in this class is close to $8$ hours? Why or why not? 

The figure displays computer output from Data Desk for data on the amount spent by $50$ grocery shoppers. 

(a) What would you guess is the shape of the distribution based only on the computer output? Explain.

 (b) Interpret the value of the standard deviation. 

(c) Are there any outliers? Justify your answer. 

C-sections Do male doctors perform more cesarean sections (C-sections) than female doctors? A study in Switzerland examined the number of cesarean sections (surgical deliveries of babies) performed in a $100$. Section $1.3$ Describing Quantitative Data with Numbers $73$ year by samples of male and female doctors. Here are summary statistics for the two distributions: 

(a) Based on the computer output, which distribution would you guess has a more symmetrical shape? Explain. 

(b) Explain how the IQRs of these two distributions can be so similar even though the standard deviations are quite different. 

(c) Does it appear that males perform more C-sections? Justify your answer 

The IQR Is the interquartile range a resistant measure of spread? Give an example of a small data set that supports your answer. 

Which of the distributions shown has a larger standard deviation? Justify your answer. 

This is a standard deviation contest. You must choose four numbers from the whole numbers $0$ to $10$, with repeats allowed. 

(a) Choose four numbers that have the smallest possible standard deviation. 

(b) Choose four numbers that have the largest possible standard deviation.

(c) Is more than one choice possible in either (a) or (b)? Explain. 

What do they measure? For each of the following summary statistics, decide (i) whether it could be used to measure center or spread and (ii) whether it is resistant. 

(a) $\left(\frac{Q_1+Q_3}{2}\right)$

(b) $\left(\frac{max-min}{2}\right)$

Here are the scores on the Survey of Study Habits and Attitudes (SSHA) for 18 first-year college women: 

and for 20 first-year college men: 

Do these data support the belief that women have better study habits and attitudes toward learning than men? (Note that high scores indicate good study habits and attitudes toward learning.) Follow the four-step process. 

Researchers from Amherst College studied the relationship between varieties of the tropical flower Heliconia on the island of Dominica and the different species of hummingbirds that fertilize the flowers.$45$ Over time, the researchers believe, the lengths of the flowers and the forms of the hummingbirds’ beaks have evolved to match each other. If that is true, flower varieties fertilized by different hummingbird species should have distinct distributions of length. The table below gives length measurements (in millimeters) for samples of three varieties of Heliconia, each fertilized by a different species of hummingbird. Do these data support the researchers’ belief? Follow the four-step process 

If a distribution is skewed to the right with no outliers, 

(a) mean $<$ median.                   (d) mean > median.

(b) mean $\approx$median.                   (e) We can’t tell without examining the data. 

(c) mean $=$ median. 

You have data on the weights in grams of $5$ baby pythons. The mean weight is 31.8 and the standard deviation of the weights is $2.39$. The correct units for the standard deviation are

(a) no units—it’s just a number. 

(b) grams.

(c) grams squared.

(d) pythons.

(e) pythons squared